关于一阶常微分方程的积分因子求解问题

一阶微分方程M(x,y)dx+N(x,y)dy=0不是全微分方程时,寻找它的积分因子成为求解方程的关键,但又是比较棘手的问题。针对这一情况,本文通过对方程的积分因子存在的充要条件定理的证明,利用定理结论求解积分因子,进而求出其通解,是一种行之有效又直观方便的方法,从而达到化难为易的目的,而且定理结论具有一般性,可以进行推广,使求积分因子时不再盲目,变得有规可循。

Solution of Integral Factor for the First Order Ordinary Differential Equation

As the first order differential equation M(x,y)dx + N(x,y)dy = 0 is not an exact differential equation,finding its integral factor becomes the key to solve the equation,which is a tough issue.In response to this situation,the purpose of this paper is to find its general solution through proving theorem of existing sufficient and necessary condition of integral factor for the equation,and using theorem conclusion to solve integral factor.This is an effective,intuitive and convenient way to achieve the purpose of turning difficult to easy.Furthermore,as the conclusion of theorem is general,it can be popularized to make solving integral factor no longer blind,but rule-based.